Extrinsic analysis on manifolds is computationally faster than intrinsic analysis with applications to quality control by machinea vision

In our technological era, non-Euclidean data abound, especially because of advances in digital imaging. Patrangenaru ('Asymptotic statistics on manifolds', PhD Dissertation, 1998) introduced extrinsic and intrinsic means on manifolds, as location parameters for non-Euclidean data. A large sample nonparametric theory of inference on manifolds was developed by Bhattacharya and Patrangenaru (J. Stat. Plann. Inferr., 108, 23-35, 2002; Ann. Statist., 31, 1-29, 2003; Ann. Statist., 33, 1211-1245, 2005). A flurry of papers in computer vision, statistical learning, pattern recognition, medical imaging, and other computational intensive applied areas using these concepts followed. While pursuing such location parameters in various instances of data analysis on manifolds, scientists are using intrinsic means, almost without exception. In this paper, we point out that there is no unique intrinsic analysis because the latter depends on the choice of the Riemannian metric on the manifold, and in dimension two or higher, there are infinitely such nonisometric choices. Also, using John Nash's celebrated isometric embedding theorem and an equivariant version, we show that for each intrinsic analysis there is an extrinsic counterpart that is computationally faster and give some concrete examples in shape and image analysis. The computational speed is important, especially in automated industrial processes. In this paper, we mention two potential applications in the industry and give a detailed presentation of one such application, for quality control in a manufacturing process via 3D projective shape analysis from multiple digital camera images.